Universal Coefficient Theorems and assembly maps in KK-theory

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1 Universal Coefficient Theorems and assembly maps in KK-theory Ralf Meyer Abstract. We introduce equivariant Kasparov theory using its universal property and construct the Baum Connes assembly map by localising the Kasparov category at a suitable subcategory. Then we explain a general machinery to construct derived functors and spectral sequences in triangulated categories. This produces various generalisations of the Rosenberg Schochet Universal Coefficient Theorem. Mathematics Subject Classification (2000). Primary 19K35; Secondary 46L80. Keywords. K-theory, KK-theory, Baum Connes assembly map, triangulated category, Universal Coefficient Theorem Contents 1 Introduction 2 I Kasparov theory and Baum Connes conjecture 3 2 Kasparov theory via its universal property Some basic homotopy theory Morita Rieffel equivalence and stable isomorphism C -stable functors Exactness properties Definition of Kasparov theory Extending functors and identities to KK G Triangulated category structure Subcategories in KK G Compactly induced actions Two simple examples II Homological algebra 20

2 2 Ralf Meyer 4 Homological ideals in triangulated categories Generalities about ideals in additive categories Examples of ideals What is a triangulated category? The universal homological functor Homological ideals in triangulated categories From homological ideals to derived functors Basic notions Exact chain complexes More homological algebra with chain complexes Projective objects Projective resolutions Derived functors Projective objects via adjointness The universal exact homological functor Derived functors in homological algebra Universal Coefficient Theorems Universal Coefficient Theorem in the hereditary case The Adams resolution Spectral sequences from the Adams resolution Introduction We may view Kasparov theory and its equivariant generalisations as categories. These categories are non-commutative analogue of (equivariant generalisations of) the stable homotopy category of spectra. These equivariant Kasparov categories can be described in two ways: Abstractly, as the universal split-exact C -stable functor on the appropriate category of C -algebras this approach is due to Cuntz and Higson [9,10,14,15]. It is useful for general constructions like the descent functor or the adjointness between induction and restriction functors (see 2.6 or [25]). Concretely, using Fredholm operators on equivariant Hilbert bimodules this is the original definition of Kasparov [16,17]. It is useful for specific constructions that use, say, geometric properties of a group to construct elements in Kasparov groups. We mainly treat Kasparov theory as a black box. We define G-equivariant Kasparov theory via its universal property and equip it with a triangulated category structure. This formalises some basic properties of the stable homotopy

3 Universal Coefficient Theorems and assembly maps in KK-theory 3 category that are needed for algebraic topology. We later apply this structure to construct spectral sequences in Kasparov theory. We use the universal property to construct the descent functor and induction and restriction functors for closed subgroups, and to verify that the latter are adjoint for open subgroups. Then we turn to the Baum Connes assembly map for a locally compact group G, which we treat as in [25]. Green s Imprimitivity Theorem suggests that we understand crossed products for compactly induced actions much better than general crossed products. We want to construct more general actions out of compactly induced actions by an analogue of the construction of CW-complexes. The notion of localising subcategory makes this idea precise. The orthogonal complement of the compactly induced actions consists of actions that are KK H -equivalent to 0 for all compact subgroups H of G. We call such actions weakly contractible. The compactly induced and weakly contractible objects together generate the whole Kasparov category. This allows us to compute the localisation of a functor at the weakly contractible objects. The general machinery of localisation yields the Baum Connes assembly map µ : K top (G, A) K (G r A) when we apply it to the functor A K (G r A). Roughly speaking, this means that A K top (G, A) is the best possible approximation to K (G r A) that vanishes for weakly contractible objects. The above statements involve functors and the Baum Connes assembly map with coefficients. The above approach only works if we study this generalisation right away. The groups K top (G, A) are supposed to be computable by topological methods. We present one approach to make this precise that works completely within equivariant Kasparov theory and is a special case of a very general machinery for constructing spectral sequences. We carry over notions from homological algebra like exact chain complexes and projective objects to our category and use them to define derived functors (see [26]). The derived functors of K (G r A) and (G, A) agree and form the E 2 -term of a spectral sequence that converges towards K top (G, A). Many other spectral sequences like the Adams spectral sequence in topology can be constructed with the same machinery. In simple special cases, the spectral sequence degenerates to an exact sequence. The Universal Coefficient Theorem by Rosenberg Schochet in [33] and the Pimsner Voiculescu exact sequence are special cases of this machinery. K top

4 4 Part I Kasparov theory and Baum Connes conjecture 2. Kasparov theory via its universal property This section is mostly taken from [24], where more details can be found. Let G be a locally compact group. Definition 1. A G-C -algebra is a C -algebra with a strongly continuous representation of G by -automorphisms. Let G-C alg be the category of G-C -algebras; its objects are G-C -algebras and its morphisms A B are the G-equivariant -homomorphisms A B; we sometimes denote this morphism set by Hom G (A, B). A C -algebra is separable if it has a countable dense subset. We often restrict attention to the full subcategory G-C sep G-C alg of separable G-C -algebras. Homology theories for C -algebras are usually required to be homotopy invariant, stable, and exact in a suitable sense. We can characterise G-equivariant Kasparov theory as the universal functor on G-C sep with these properties, in the following sense. Definition 2. Let P be a property for functors defined on G-C sep. A universal functor with P is a functor u: G-C sep U P (G-C sep) such that F u has P for each functor F : U P (G-C sep) C; any functor F : G-C sep C with P factors uniquely as F = F u for some functor F : U P (G-C sep) C. Of course, such a functor need not exist. If it does, then it restricts to a bijection between objects of G-C sep and U P (G-C sep). Hence we can completely describe it by the sets of morphisms U P (A, B) from A to B in U P (G-C sep) and the maps G-C sep(a, B) U P (A, B) for A, B G-C sep. The universal property means that for any functor F : G-C sep C with P there is a unique functorial way to extend the maps Hom G (A, B) C ( F(A), F(B) ) to U P (A, B) Some basic homotopy theory. We define cylinders, cones, and suspensions of objects and mapping cones and mapping cylinders of morphisms in G-C alg. Then we define homotopy invariance for functors. Mapping cones will be used later to introduce the triangulated category structure on Kasparov theory.

5 5 Notation 3. Let A be a G-C -algebra. We define the cylinder, cone, and suspension over A by ( Cyl(A) := C([0, 1], A), Sus(A) := C 0 [0, 1] \ {0, 1}, A) = C0 (S 1, A), ( Cone(A) := C 0 [0, 1] \ {0}, A), If A = C 0 (X) for a pointed compact space, then the cylinder, cone, and suspension of A are C 0 (Y ) with Y equal to the usual cylinder [0, 1] + X, cone [0, 1] X, or suspension S 1 X, respectively; here [0, 1] has the base point 0. Definition 4. Let f : A B be a morphism in G-C alg. The mapping cylinder Cyl(f) and the mapping cone Cone(f) of f are the limits of the diagrams A f B ev1 Cyl(B), A f B ev1 Cone(B) in G-C alg. More concretely, Cone(f) = { (a, b) A C 0 ( (0, 1], B ) f(a) = b(1) }, Cyl(f) = { (a, b) A C ( [0, 1], B ) f(a) = b(1) }. If f : X Y is a morphism of pointed compact spaces, then the mapping cone and mapping cylinder of the induced -homomorphism C 0 (f): C 0 (Y ) C 0 (X) agree with C 0 ( Cyl(f) ) and C0 ( Cone(f) ), respectively. The familiar maps relating mapping cones and cylinders to cones and suspensions continue to exist in our case. For any morphism f : A B in G-C alg, we get a morphism of extensions Sus(B) Cone(f) A Cone(B) Cyl(f) A The bottom extension splits and the maps A Cyl(f) are inverse to each other up to homotopy. The composite map Cone(f) A B factors through Cone(id B ) = Cone(B) and hence is homotopic to the zero map. Definition 5. Let f 0, f 1 : A B be two parallel morphisms in G-C alg. We write f 0 f 1 and call f 0 and f 1 homotopic if there is a morphism f : A Cyl(B) with ev t f = f t for t = 0, 1. A functor F : G-C alg C is called homotopy invariant if f 0 f 1 implies F(f 0 ) = F(f 1 ). It is easy to check that homotopy is an equivalence relation on Hom G (A, B). We let [A, B] be the set of equivalence classes. The composition of morphisms in G-C alg descends to maps [B, C] [A, B] [A, C], ( [f], [g] ) [f g],

6 6 that is, f 1 f 2 and g 1 g 2 implies f 1 f 2 g 1 g 2. Thus the sets [A, B] form the morphism sets of a category, called the homotopy category of G-C -algebras. A functor is homotopy invariant if and only if it descends to the homotopy category. Other characterisations of homotopy invariance are listed in [24, 3.1]. Of course, our notion of homotopy restricts to the usual one for pointed compact spaces or to proper homotopy for locally compact spaces Morita Rieffel equivalence and stable isomorphism. One of the basic ideas of non-commutative geometry is that G C 0 (X) (or G r C 0 (X)) should be a substitute for the quotient space G\X, which may have bad singularities. In the special case of a free and proper G-space X, we expect that G C 0 (X) and C 0 (G\X) are equivalent in a suitable sense. Already the simplest possible case X = G shows that we cannot expect an isomorphism here because G C 0 (G) = G r C 0 (G) = K(L 2 G). The right notion of equivalence is a C -version of Morita equivalence introduced by Marc A. Rieffel ([30 32]); therefore, we call it Morita Rieffel equivalence. The definition of Morita Rieffel equivalence involves Hilbert modules over C -algebras and the C -algebras of compact operators on them; these notions are crucial for Kasparov theory as well. We refer to [19] for the definition and a discussion of their basic properties. Definition 6. Two G-C -algebras A and B are called Morita Rieffel equivalent if there are a full G-equivariant Hilbert B-module E and a G-equivariant -isomorphism K(E) = A. It is possible (and desirable) to express this definition more symmetrically: E is an A, B-bimodule with two inner products taking values in A and B, satisfying various conditions (see also [30]). Two Morita Rieffel equivalent G-C -algebras hve equivalent categories of G-equivariant Hilbert modules via E B. The converse is not so clear. Example 7. The following is a more intricate example of a Morita Rieffel equivalence. Let Γ and P be two subgroups of a locally compact group G. Then Γ acts on G/P by left translation and P acts on Γ\G by right translation. The corresponding orbit space is the double coset space Γ\G/P. Both Γ C 0 (G/P) and P C 0 (Γ\G) are non-commutative models for this double coset space. They are indeed Morita Rieffel equivalent; the bimodule that implements the equivalence is a suitable completion of C c (G). These examples suggest that Morita Rieffel equivalent C -algebras are different ways to describe the same non-commutative space. Therefore, we expect that reasonable functors on C alg should not distinguish between Morita Rieffel equivalent C -algebras. Definition 8. Two G-C -algebras A and B are called stably isomorphic if there is a G-equivariant -isomorphism A K(H G ) = B K(H G ), where H G := L 2 (G N)

7 7 is the direct sum of countably many copies of the regular representation of G; we let G act on K(H G ) by conjugation, of course. The following technical condition is often needed in connection with Morita Rieffel equivalence. Definition 9. A C -algebra is called σ-unital if it has a countable approximate identity or, equivalently, contains a strictly positive element. All separable C -algebras and all unital C -algebras are σ-unital; the algebra K(H) is σ-unital if and only if H is separable. Theorem 10 ([7]). σ-unital G-C -algebras are G-equivariantly Morita Rieffel equivalent if and only if they are stably isomorphic. In the non-equivariant case, this theorem is due to Brown Green Rieffel ([7]). A simpler proof that carries over to the equivariant case appeared in [27] C -stable functors. The definition of C -stability is more intuitive in the non-equivariant case: Definition 11. Fix a rank-one projection p K(l 2 N). The resulting embedding A A K(l 2 N), a a p, is called a corner embedding of A. A functor F : C alg C is called C -stable if any corner embedding induces an isomorphism F(A) = F ( A K(l 2 N) ). The correct equivariant generalisation is the following: Definition 12. A functor F : G-C alg C is called C -stable if the canonical embeddings H 1 H 1 H 2 H 2 induce isomorphisms F ( A K(H 1 ) ) = F ( A K(H 1 H 2 ) ) = F ( A K(H 2 ) ) for all non-zero G-Hilbert spaces H 1 and H 2. Of course, it suffices to require F ( A K(H 1 ) ) = F ( A K(H 1 H 2 ) ). It is not hard to check that Definitions 11 and 12 are equivalent for trivial G. Our next goal is to describe the universal C -stable functor. We abbreviate A K := K(L 2 G) A. Definition 13. A correspondence from A to B (or A B) is a G-equivariant Hilbert B K -module E together with a G-equivariant essential (or non-degenerate) -homomorphism f : A K K(E). Given correspondences E from A to B and F from B to C, their composition is the correspondence from A to C with underlying Hilbert module E BK F and map A K K(E) K(E BK F), where the last map sends T T 1; this yields compact operators because B K maps to K(F). See [19] for the definition of the relevant completed tensor product of Hilbert modules.

8 8 The composition of correspondences is only defined up to isomorphism. It is associative and the identity maps A A = K(A) act as unit elements, so that we get a category Corr G whose morphisms are the isomorphism classes of correspondences. Any -homomorphism ϕ: A B yields a correspondence: let E be the right ideal ϕ(a K ) B K in B K, viewed as a Hilbert B-module, and let ϕ(a) b = ϕ(a) b; this restricts to a compact operator on E. This defines a canonical functor : G-C alg Corr G. Proposition 14. The functor : G-C alg Corr G is the universal C -stable functor on G-C alg; that is, it is C -stable, and any other such functor factors uniquely through. Proof. First we sketch the proof in the non-equivariant case. First we must verify that is C -stable. The Morita Rieffel equivalence between K(l 2 N) A = K ( l 2 (N, A) ) and A is implemented by the Hilbert module l 2 (N, A), which yields a correspondence ( id, l 2 (N, A) ) from K(l 2 N) A to A; this is inverse to the correspondence induced by a corner embedding A K(l 2 N) A. A Hilbert B-module E with an essential -homomorphism A K(E) is countably generated because A is assumed σ-unital. Kasparov s Stabilisation Theorem yields an isometric embedding E l 2 (N, B). Hence we get -homomorphisms A K(l 2 N) B B. This diagram induces a map F(A) F(K(l 2 N) B) = F(B) for any stable functor F. Now we should check that this well-defines a functor F : Corr G C with F = F, and that this yields the only such functor. We omit these computations. The generalisation to the equivariant case uses the crucial property of the left regular representation that L 2 (G) H = L 2 (G N) for any countably infinitedimensional G-Hilbert space H. Since we replace A and B by A K and B K in the definition of correspondence right away, we can use this to repair a possible lack of G-equivariance; similar ideas appear in [22]. Example 15. Let u be a G-invariant multiplier of B. Then the identity map and the inner automorphism B B, b ubu, defined by u define isomorphic correspondences B B (via u). Hence inner automorphims act trivially on C -stable functors. Actually, this is one of the computations that we have omitted in the proof above; the argument can be found in [11] Exactness properties. Definition 16. A diagram I E Q in G-C alg is an extension if it is isomorphic to the canonical diagram I A A/I for some G-invariant ideal I in a G-C -algebra A. We write I E Q to denote extensions. A section for an extension I i E p Q (17) in G-C alg is a map (of sets) Q E with p s = id Q. We call (17) split if there is a section that is a G-equivariant -homomorphism. We call (17) G-equivariantly cp-split if there is a G-equivariant, completely positive, contractive, linear section.

9 9 Sections are also often called lifts, liftings, or splittings. Definition 18. A functor F on G-C alg is split-exact if, for any split extension K i E p Q with section s: Q E, the map ( F(i), F(s) ) : F(K) F(Q) F(E) is invertible. Split-exactness is useful because of the following construction of Joachim Cuntz ([9]). Let B E be a G-invariant ideal and let f +, f : A E be G-equivariant -homomorphisms with f + (a) f (a) B for all a A. Equivalently, f + and f both lift the same morphism f : A E/B. The data (A, f +, f, E, B) is called a quasi-homomorphism from A to B. Pulling back the extension B E E/B along f, we get an extension B E A with two sections f +, f : A E. The split-exactness of F shows that F(B) F(E ) F(A) is a split extension in C. Since both F(f ) and F(f +) are sections for it, we get a map F(f +) F(f ): F(A) F(B). Thus a quasi-homomorphism induces a map F(A) F(B) if F is split-exact. The formal properties of this construction are summarised in [11]. Given a C -algebra A, there is a universal quasi-homomorphism out of A. Let Q(A) := A A be the free product of two copies of A and let π A : Q(A) A be the folding homomorphism that restricts to id A on both factors. Let q(a) be its kernel. The two canonical embeddings A A A are sections for the folding homomorphism. Hence we get a quasi-homomorphism A Q(A) q(a). The universal property of the free product shows that any quasi-homomorphism yields a G-equivariant -homomorphism q(a) B. Theorem 19. Functors that are C -stable and split-exact are automatically homotopy invariant. This is a deep result of Nigel Higson ([15]); a simple proof can be found in [11]. Besides basic properties of quasi-homomorphisms, it only uses that inner endomorphisms act identically on C -stable functors. Definition 20. We call F exact if F(K) F(E) F(Q) is exact (at F(E)) for any extension K E Q in S. More generally, given a class E of extensions in S like, say, the class of equivariantly cp-split extensions, we define exactness for extensions in E. Most functors we are interested in satisfy homotopy invariance and Bott periodicity, and these two properties prevent a functor from being exact in the stronger sense of being left or right exact. This explains why our notion of exactness is much weaker than usual in homological algebra. It is reasonable to require that a functor be part of a homology theory, that is, a sequence of functors (F n ) n Z together with natural long exact sequences for all extensions. We do not require this because this additional information tends to be hard to get a priori but often comes for free a posteriori:

10 10 Proposition 21. Suppose that F is homotopy invariant and exact (or exact for equivariantly cp-split extensions). Then F has long exact sequences of the form F ( Sus(K) ) F ( Sus(E) ) F ( Sus(Q) ) F(K) F(E) F(Q) for any (equivariantly cp-split) extension K E Q. In particular, F is splitexact. See 21.4 in [4] for the proof. Together with Bott periodicity, this yields long exact sequences that extend towards ± in both directions, showing that an exact homotopy invariant functor that satisfies Bott periodicity is part of a homology theory in a canonical way Definition of Kasparov theory. Kasparov theory associates to two Z/2-graded C -algebras an Abelian group KK G 0 (A, B); this is a vast generalisation of K-theory and K-homology. The most remarkable feature of this theory is an associative product on KK called Kasparov product, which generalises various known product constructions in K-theory and K-homology. We do not discuss KK G for Z/2-graded G-C -algebras here because it does not fit so well with the universal property approach. Fix a locally compact group G. The Kasparov groups KK G 0 (A, B) for A, B G-C sep form morphisms sets A B of a category, which we denote by KK G ; the composition in KK G is the Kasparov product. The categories G-C sep and KK G have the same objects. We have a canonical functor that acts identically on objects. KK: G-C sep KK G Theorem 22. The functor KK G : G-C sep KK G is the universal split-exact C -stable functor; in particular, KK G is an additive category. In addition, KK G also has the following properties and is, therefore, universal among functors with some of these extra properties: KK G is homotopy invariant; exact for G-equivariantly cp-split extensions; satisfies Bott periodicity, that is, in KK G there are natural isomorphisms Sus 2 (A) = A for all A KK G. Definition 23. A G-equivariant -homomorphism f : A B is called a KK G - equivalence if KK(f) is invertible in KK G. Corollary 24. Let F : G-C sep C be split-exact and C -stable. Then F factors uniquely through KK G, is homotopy invariant, and satisfies Bott periodicity. A KK G -equivalence A B induces an isomorphism F(A) F(B).

11 11 We will take the universal property of Theorem 22 as a definition of KK G and thus of the groups KK G 0 (A, B). We also let KK G n (A, B) := KK G( A, Sus n (B) ) ; since the Bott periodicity isomorphism identifies KK G 2 = KK G 0, this yields a Z/2-graded theory. By the universal property, K-theory descends to a functor on KK, that is, we get canonical maps KK 0 (A, B) Hom ( K (A), K (B) ) for all separable C -algebras A, B, where the right hand side denotes gradingpreserving group homomorphisms. For A = C, this yields a map KK 0 (C, B) Hom ( Z, K 0 (B) ) = K0 (B). Using suspensions, we also get a corresponding map KK 1 (C, B) K 1 (B). Theorem 25. The maps KK (C, B) K (B) constructed above are isomorphisms for all B C sep. Thus Kasparov theory is a bivariant generalisation of K-theory. Roughly speaking, KK (A, B) is the place where maps between K-theory groups live. Most constructions of such maps, say, in index theory can in fact be improved to yield elements of KK (A, B). One reason for this is the Universal Coefficient Theorem (UCT) by Rosenberg and Schochet [33], which computes KK (A, B) from K (A) and K (B) for many C -algebras A, B. If A satisfies the UCT, then any group homomorphism K (A) K (B) lifts to an element of KK (A, B) of the same parity. With our definition, it is not obvious how to construct elements in KK G 0 (A, B). The only source we know so far are G-equivariant -homomorphisms. Another important source are extensions, more precisely, equivariantly cp-split extensions. Any such extension I E Q yields a class in KK G 1 (Q, I) = KK G 0 (Sus(Q), I) = KK G ( ) 0 Q, Sus(I). Conversely, any element in KK G 1 (Q, I) comes from an extension in this fashion in a rather transparent way. Thus it may seem that we can understand all of Kasparov theory from an abstract, category theoretic point of view. But this is not the case. To get a category, we must compose extensions; this leads to extensions of higher length. If we allow such higher-length extensions, we can easily construct a category that is isomorphic to Kasparov theory; this generalisation still works for more general algebras than C -algebras (see [11]) because it does not involves any difficult analysis any more. But such a setup offers no help to compute products. Here computing products means identifying them with other simple things like, say, the identity morphism. This is why the more concrete approach to Kasparov theory is still necessary for the interesting applications of the theory. In connection with the Baum Connes conjecture, our abstract approach allows us to formulate it and analyse its consequences. But to verify it, say, for amenable groups, we must show that a certain morphism in KK G is invertible.

12 12 This involves constructing its inverse and checking that the two Kasparov products in both order are 1. These computations require the concrete description of Kasparov theory that we omit here. We merely refer to [4] for a detailed treatment Extending functors and identities to KK G. We use the universal property to extend functors from G-C alg to KK G and check identities in KK G without computing Kasparov products. As our first example, consider the full and reduced crossed product functors G r, G : G-C alg C alg. Proposition 26. These two functors extend to functors called descent functors. G r, G : KK G KK Kasparov constructs these functors directly using the concrete description of Kasparov cycles. This requires a certain amount of work; in particular, checking functoriality involves knowing how to compute Kasparov products. Proof. We only write down the argument for reduced crossed products, the other case is similar. It is well-known that G r ( A K(H) ) = (G r A) K(H) for any G-Hilbert space H. Therefore, the composite functor G-C sep G r C sep KK KK is C -stable. This functor is split-exact as well (we omit the proof). Now the universal property provides an extension to a functor KK G KK. Similarly, we get functors A min, A max : KK G KK G for any G-C -algebra A. Since these extensions are natural, we even get bifunctors min, max : KK G KK G KK G. For the Baum Connes assembly map, we need the induction functors Ind G H : KKH KK G for closed subgroups H G. For a finite group H, Ind G H(A) is the H-fixed point algebra of C 0 (G, A), where H acts by h f(g) = α h ( f(gh) ). For infinite H, we have Ind G H(A) = {f C b (G, A) α h f(gh) = f(g) for all g G, h H, and gh f(g) is C 0 };

13 13 the group G acts by translations on the left. This construction is clearly functorial for equivariant -homomorphisms. Furthermore, it commutes with C -stabilisations and maps split extensions again to split extensions. Therefore, the same argument as above allows us to extend it to a functor Ind G H : KKH KK G The following examples are more trivial. Let τ : C alg G-C alg equip a C -algebra with the trivial G-action; it extends to a functor τ : KK KK G. The restriction functors Res H G : KK G KK H for closed subgroups H G are defined by forgetting part of the equivariance. The universal property also allows us to prove identities between functors. For instance, Green s Imprimitivity Theorem provides -isomorphisms G Ind G H(A) M H A, G r Ind G H(A) M H r A (27) for any H-C -algebra A. This is proved by completing C 0 (G, A) to an imprimitivity bimodule for both C -algebras. This equivalence is clearly natural for H-equivariant -homomorphisms. Since all functors involved are C -stable and split exact, the uniqueness part of the universal property of KK H shows that the KK-equivalences G Ind G H (A) = H A and G r Ind G H (A) = H r A are natural for morphisms in KK H. That is, the diagram G r Ind G H (A 1) G rind G H (f) G r Ind G H (A 2) = H r A 1 H rf = H r A 2 in KK commutes for any f KK H 0 (A 1, A 2 ). More examples of this kind are discussed in 4.1 of [24]. We can also prove adjointness relations in Kasparov theory in an abstract way by constructing the unit and counit of the adjunction. An important example is the adjointness between induction and restriction functors (see also 3.2 of [25]). Proposition 28. Let H G be a closed subgroup. If H is open, then we have natural isomorphisms KK G (Ind G H A, B) = KK H (A, Res H G B) (29) for all A H-C alg, B G-C alg. If H G is cocompact, then we have natural isomorphisms KK G (A, Ind G H B) = KK H (Res H G A, B) (30) for all A G-C alg, B H-C alg.

14 14 Proof. We will not use (30) later and therefore only prove (29). We must construct natural elements α A KK G 0 (IndG H ResH G A, A), β B KK H 0 (B, ResH G IndG H B) that satisfy the conditions for unit and counit of adjunction ([21]). We have a natural G-equivariant -isomorphism Ind G H Res G H(A) = C 0 (G/H) A for any G-C -algebra A. Since H is open in G, the homogeneous space G/H is discrete. We represent C 0 (G/H) on the Hilbert space l 2 (G/H) by pointwise multiplication operators. This is G-equivariant for the representation of G on l 2 (G/H) by left translations. Thus we get a correspondence from Ind G H ResH G (A) to A, which yields α A KK G 0 (Ind G H Res H G(A), A) because KK G is C -stable. For any H-C -algebra B, we may embed B in Res H G Ind G H(B) as the subalgebra of functions supported on the single coset H. This embedding is H-equivariant and provides β B KK H 0 (B, Res H G Ind G H B). Now we have to check that the following two composite maps are identity morphisms in KK G and KK H, respectively: Ind G H (B) IndG H (βb) Ind G H ResH G IndG H (B) α Ind G H (B) Ind G H (B) Res H G A β Res H G A Res H G IndG H ResH G (A) ResH G αa Res H G A This yields the desired adjointness by a general argument from category theory (see [21]). In fact, both composites are already equal to the identity as correspondences. Hence we need no knowledge of Kasparov theory except for its C -stability to prove (29). The following example is discussed in detail in 4.1 of [24]. If G is compact, then the trivial action functor τ : KK KK G is left adjoint to G = G r, that is, we have natural isomorphisms KK G (τ(a), B) = KK (A, G B). (31) This is also known as the Green Julg Theorem. For A = C, it specialises to a natural isomorphism K G (B) = K (G B) Triangulated category structure. We can turn KK G into a triangulated category by extending standard constructions for topological spaces (see [25]). But some arrows change direction because the functor C 0 from spaces to C -algebras is contravariant. We have already observed that KK G is additive. The suspension automorphism is Σ 1 (A) := Sus(A). Since Sus 2 (A) = A in KK G by Bott periodicity, we have Σ = Σ 1. Thus we do not need formal desuspensions as for the stable homotopy category. Definition 32. A triangle A B C ΣA in KK G is called exact if it is isomorphic as a triangle to the mapping cone triangle Sus(B) Cone(f) A f B for some G-equivariant -homomorphism f.

15 15 Alternatively, we can use G-equivariantly cp-split extensions in G-C sep. Any such extension I E Q determines a class in KK G 1 (Q, I) = KK G 0 (Sus(Q), I), so that we get a triangle Sus(Q) I E Q in KK G. Such triangles are called extension triangle. A triangle in KK G is exact if and only if it is isomorphic to the extension triangle of a G-equivariantly cp-split extension. Theorem 33. With the suspension automorphism and exact triangles defined above, KK G is a triangulated category. Proof. This is proved in detail in [25]. Triangulated categories clarify the basic bookkeeping with long exact sequences. Mayer-Vietoris exact sequences and inductive limits are discussed from this point of view in [25]. More importantly, this framework sheds light on more advanced constructions like the Baum Connes assembly map. 3. Subcategories in KK G Now we turn to the construction of the Baum Connes assembly map by studying various subcategories of KK G that are related to it Compactly induced actions. Definition 34. Let G be a locally compact group. A G-C -algebra is compactly induced if it is of the form Ind G H (A) for some compact subgroup H of G and some H-C -algebra A. We let CI be the class of all G-C -algebras that are KK G - equivalent to a direct summand of i N A i with compactly induced G-C -algebras A i for i N. Equivalently, CI is the smallest class of objects in KK G that is closed under direct sums, direct summands and isomorphism and contains all compactly induced G-C -algebras. Green s Imprimitivity Theorem (27) tells us that the (reduced) crossed product for a compactly induced action Ind G H (A) is equivalent to the crossed product H r A for the compact group H. Hence we have K (G Ind G H A) = K (G r Ind G H A) = K (H A) = K H (A) by the Green Julg Theorem, compare (31). Since the computation of equivariant K-theory for compact groups is a problem of classical topology, operator algebraists can pretend that it is Somebody Else s Problem. We are more fascinated by the analytic difficulties created by crosssed products by infinite groups. For instance, it is quite hard to see which Laurent series n Z a nz n correspond to an element of C red Z = C Z or, equivalently, which of them are the Fourier series of a continuous function on the unit circle. The Baum Connes conjecture, when true, implies that such analytic difficulties do not influence the K-theory.

16 Two simple examples. It is best to explain our goals with two examples, namely, the groups R and Z. The Baum Connes conjectures for these groups hold and are equivalent to the Connes Thom isomorphism and a Pimsner Voiculescu exact sequence. Although the Baum Connes conjecture only concerns the K-theory of Cred G and, more generally, of crossed products G r A, we get much stronger statements in this case. Both R and Z are torsion-free, that is, they have no non-trivial compact subgroups. Hence the compactly induced actions are of the form C 0 (G, A) with G {R, Z} acting by translation. If A carries another action of G, then it makes no difference ( whether we let G act on C 0 (G, A) by t f(x) := f(t 1 x) or t f(x) := α t f(t 1 x) ) : both definitions yield isomorphic G-C -algebras. Theorem 35. Any R-C -algebra is KK R -equivalent to a compactly induced one. More briefly, CI = KK R. Proof. Let A be any R-C -algebra. Let R act on R by translation and extend this to an action on X = (, ] by t := for all t R. Then we get an extension of R-C -algebras C 0 (R, A) C 0 (X, A) A, where we let R act diagonally. It does not yet have an R-equivariant completely positive section, but it becomes equivariantly cp-split if we tensor with K(L 2 G). Therefore, it yields an extension triangle in KK R. The Dirac operator on C 0 (R, A) for the standard Riemannian metric on R defines a class in KK R 1 (C 0 (R), C) which we may then map to KK R 1(C 0 (R, A), A) by exterior product. This yields another cp-split extension K(L 2 R) A T A C 0 (R, A). The resulting classes in KK G ( 1 A, C0 (R, A) ) and KK G 1 (C 0 (R, A), A) are inverse to each other; this is checked by computing their Kasparov products in both orders. Thus A is KK R -equivalent to the induced R-C -algebra C 0 (R, A). Since the crossed product is functorial on Kasparov categories, this implies R A = R r A R r Sus ( C 0 (R, A) ) = Sus(K(L 2 R) A) Sus(A), where denotes KK-equivalence. Taking K-theory, we get the Connes Thom Isomorphism K (R A) = K +1 (A). For most groups, we have CI KK G. We now study the simplest case where this happens, namely, G = Z. We have seen above that C 0 (R) with the translation action of R is KK R - equivalent to C 0 (R) with trivial action. This equivalence persists if we restrict the action from R to the subgroup Z R. Hence we get a KK Z -equivalence A Sus ( C 0 (R, A) ), where n Z acts on C 0 (R, A) = C 0 (R) A by (α n f)(x) := α n ( f(x n) ). Although the Z-action on R is free and proper, the action of Z on C 0 (R, A) need not be induced from the trivial subgroup.

17 17 Theorem 36. For any Z-C -algebra A, there is an exact triangle P 1 P 0 A ΣP 1 in KK Z with compactly induced P 0 and P 1 ; more explicitly, P 0 = P 1 = C 0 (Z, A). Proof. Restriction to Z R provides a surjection C 0 (R, A) C 0 (Z, A), whose kernel may be identified with C 0 ((0, 1)) C 0 (Z, A). The resulting extension C 0 ((0, 1)) C 0 (Z, A) C 0 (R, A) C 0 (Z, A) is Z-equivariantly cp-split and hence provides an extension triangle in KK Z. Since C 0 (R, A) is KK Z -equivalent to the suspension of A, we get an exact triangle of the desired form. When we apply a homological functor KK G C such as K (Z ) to the exact triangle in Theorem 36, then we get the Pimsner Voiculescu exact sequence K 1 (A) K 0 (Z A) K 0 (A) α 1 K 1 (A) K 1 (Z A) α 1 K 0 (A). Here α : K (A) K (A) is the map induced by the automorphism α(1) of A. It is not hard to identify the boundary map for the above extension with this map. Our approach yields such exact sequences for any homological functor. Now we formulate some structural results for R and Z that have a chance to generalise to other groups. Theorem 37. Let G be R or Z. Let A 1 and A 2 be G-C -algebras and let f KK G (A 1, A 2 ). If Res G (f) KK(A 1, A 2 ) is invertible, then so is f itself. In particular, if Res G (A 1 ) = 0 in KK, then already A = 0 in KK G. Proof. We only write down the proof for G = Z; the case G = R is similar but simpler. If f were an equivariant -homomorphisms, then it would induce a morphism of extensions ( ) ( ) ( ) C 0 (0, 1) Z, A1 C 0 R, A1 C 0 Z, A1 f C 0 ( (0, 1) Z, A2 ) f C 0 ( R, A2 ) f C 0 ( Z, A2 ) and hence a morphism of triangles between the resulting extension triangles. The latter morphism still exists even if f is merely a morphism in KK Z. This can be checked directly or deduced in a routine fashion from the uniqueness part of the universal property of KK Z. If Res G (f) is invertible, then so are the induced maps C 0 ( (0, 1) Z, A1 ) C0 ( (0, 1) Z, A2 ) and C0 (Z, A 1 ) C 0 (Z, A 2 ) because C 0 (Z, A) = Ind Z Res Z (A). Hence the Five Lemma in triangulated categories shows that f itself is invertible. To get the second statement, apply the first one to the zero maps 0 A 1 0.

18 18 Definition 38. A path of G-actions (α t ) t [0,1] is continuous if its pointwise application defines a strongly continuous action of G on Cyl(A) := C([0, 1], A). Corollary 39. Let G = R or Z. If (α t ) t [0,1] is a continuous path of G-actions on A, then there is a canonical KK G -equivalence (A, α 0 ) (A, α 1 ). As a consequence, the crossed products for both actions are KK-equivalent. Proof. Equip Cyl(A) with the automorphism α. Evaluation at 0 and 1 provides elements in KK Z( Cyl(A), (A, α t ) ) that are non-equivariantly invertible because KK is homotopy invariant. Hence they are invertible in KK G by Theorem 37. Their composition yields the desired KK G -equivalence (A, α 0 ) (A, α 1 ). It is not hard to extend Theorem 37 and hence Corollary 39 to the groups R n and Z n for any n Z. With a bit more work, we could also treat solvable Lie groups. But Theorem 37 as stated above fails for finite groups: there exists a space X and two homotopic actions α 0, α 1 of Z/2 on X for which K Z/2 (X, α t) are different for t = 0, 1. Reversing the argument in the proof of Corollary 39, this provides the desired counterexample. Less complicated counterexamples can be constructed where A is a UHF C -algebra. Such counterexamples force us to amend our question: Suppose Res H G(A) = 0 for all compact subgroups H G. Does it follow that A = 0 in KK G? Or at least that K (G r A) = 0? It is shown in [25] that the second question has a positive answer if and only if the Baum Connes conjecture holds for G with arbitrary coefficients. For many groups for which we know the Baum Connes conjecture with coefficients, we also know that the first question has a positive answer. But the first question can only have a positive answer if the group is K-amenable, that is, if reduced and full crossed products have the same K-theory. The Lie group Sp(n, 1) and its cocompact subgroups are examples where we know the Baum Connes conjecture with coefficients although the group is not K-amenable. Definition 40. A G-C -algebra A is called weakly contractible if Res H G (A) = 0 for all compact subgroups H G. Let CC be the class of weakly contractible objects. A morphism f KK G (A 1, A 2 ) is called a weak equivalence if Res H G(f) is invertible for all compact subgroups H G. Recall that any f KK G (A 1, A 2 ) is part of an exact triangle A 1 A 2 C ΣA 1 in KK G. We have C CC if and only if f is a weak equivalence. Hence our two questions above are equivalent to: Are all weak equivalences invertible in KK G? Do they at least act invertibly on K (G r )? The second question is equivalent to the Baum Connes conjecture. Suppose now that G is discrete. Then any subgroup is open, so that the adjointness isomorphism (29) always applies. It asserts that the subcategories CI and CC are orthogonal, that is, KK G (A, B) = 0 if A CI, B CC. Even more,

19 19 if KK G (A, B) = 0 for all A CI, then it follows that B CC. A more involved argument in [25] extends these observations to all locally compact groups G. Definition 41. Let CI be the smallest full subcategory of KK G that contains CI and is closed under suspensions, (countable) direct sums, and exact triangles. We may think of objects of CI as generalised CW-complexes that are built out of the cells in CI. Theorem 42. The pair of subcategories ( CI, CC) is complementary in the following sense (see [25]): KK G (P, N) = 0 if P CI, N CC; for any A KK G, there is an exact triangle P A N ΣP with P CI, N CC. Moreover, the exact triangle P A N ΣP above is unique up to a canonical isomorphism and depends functorially on A, and the ensuing functors A P(A), A N(A) are exact functors on KK G. Proof. The orthogonality of CI and CC follows easily from the orthogonality of CI and CC. The existence of an exact triangle decomposition is more difficult. The proof in [25] reduces this to the special case A = C. A more elementary construction of this exact triangle is explained in [11]. Theorem 42 asserts that CI and CC together generate all of KK G. This is why the vanishing of K (G r A) for A CC is so useful: it allows us to replace an arbitrary object by one in CI. The latter is built out of objects in CI. We have already agreed that the computation of K (G r A) for A CI is Somebody Else s Problems. Once we understand a mechanism for decomposing objects of CI into objects of CI, the computation of K (G r A) for A CI becomes a purely topological affair and hence Somebody Else s Problem as well. For the groups Z n and R n, the subcategory CC is trivial, so that Theorem 42 simply asserts that KK G = CI is generated by the compactly induced actions. More generally, this is the case for all amenable groups; the proof of the Baum Connes conjecture by Higson and Kasparov for such groups also yields this stronger assertion (see [25]). Definition 43. Let F : KK G C be a functor. Its localisation at CC (or at the weak equivalences) is the functor LF := F P : KK G CI KK G C, where we use the functors P : KK G CI and N : KK G CC that are part of a natural exact triangle P(A) A N(A) ΣP(A). The natural transformation P(A) A furnishes a natural transformation LF(A) F(A). If F is homological or exact, then F N(A) is the obstruction to invertibility of this map.

20 20 The localisation LF can be characterised by a universal property. First of all, it vanishes on CC because P(A) = 0 whenever A CC. If F is another functor with this property, then any natural transformation F F factors uniquely through LF F. This universal property characterises LF uniquely up to natural isomorphism of functors. Theorem 44. The natural transformation LF(A) F(A) for F(A) := K (G r A) is equivalent to the Baum Connes assembly map. That is, there is a natural isomorphism LF(A) = K top (G, A) compatible with the maps to F(A). Proof. It is known (but not obvious) that K top (G, A) vanishes for CC and that the Baum Connes assembly map is an isomorphism for coefficients in CI. These two facts together imply the result. The Baum Connes conjecture asserts that the assembly map LF(A) F(A) is invertible for all A if F(A) := K (G r A). This follows if CI = KK G, of course. In particular, the Baum Connes conjecture is trivial if G itself is compact. Part II Homological algebra It is well-known that many basic constructions from homotopy theory extend to categories of C -algebras. As we argued in [25], the framework of triangulated categories is ideal for this purpose. The notion of triangulated category was introduced by Jean-Louis Verdier to formalise the properties of the derived category of an Abelian category. Stable homotopy theory provides further classical examples of triangulated categories. The triangulated category structure encodes basic information about manipulations with long exact sequences and (total) derived functors. The main point of [25] is that the domain of the Baum Connes assembly map is the total left derived functor of the functor that maps a G-C -algebra A to K (G r A). Projective resolutions are among the most fundamental concepts in homological algebra; several others like derived functors are based on it. Projective resolutions seem to live in the underlying Abelian category and not in its derived category. This is why total derived functor make more sense in triangulated categories than the derived functors themselves. Nevertheless, we can define derived functors in triangulated categories and far more general categories. This goes back to S. Eilenberg and J. C. Moore ([12]). We learned about this theory in articles by Apostolos Beligiannis ([3]) and J. Daniel Christensen ([8]). Homological algebra in non-abelian categories is always relative, that is, we need additional structure to get started. This is useful because we may fit the additional data to our needs. In a triangulated category T, there are several kinds of additional data that yield equivalent theories; following [8], we use an ideal in T. We only consider ideals I T of the form I(A, B) := {x T(A, B) F(x) = 0}

21 21 for a stable homological functor F : T C into a stable Abelian category C. Here stable means that C carries a suspension automorphism and that F intertwines the suspension automorphisms on T and C, and homological means that exact triangles yield exact sequences. Ideals of this form are called homological ideals. A basic example is the ideal in the Kasparov category KK defined by I K (A, B) := {f KK(A, B) 0 = K (f): K (A) K (B)}. (45) For a locally compact group G and a (suitable) family of subgroups F, we define the homological ideal VC F (A, B) := {f KK G (A, B) Res H G (f) = 0 in KKH (A, B) for all H F}. (46) If F is the family of compact subgroups, then VC F is related to the Baum Connes assembly map ([25]). Of course, there are analogous ideals in more classical categories of (spectra of) G-CW-complexes. All these examples can be analysed using the machinery we explain; but we only carry this out in some cases. We use an ideal I to carry over various notions from homological algebra to our triangulated category T. In order to see what they mean in examples, we characterise them using a stable homological functor F : T C with kerf = I. This is often easy. For instance, a chain complex with entries in T is I-exact if and only if F maps it to an exact chain complex in the Abelian category C, and a morphism in T is an I-epimorphism if and only if F maps it to an epimorphism. Here we may take any functor F with kerf = I. But the most crucial notions like projective objects and resolutions require a more careful choice of the functor F. Here we need the universal I-exact functor, which is a stable homological functor F with kerf = I such that any other such functor factors uniquely through F (up to natural equivalence). The universal I-exact functor and its applications are due to Apostolos Beligiannis ([3]). If F : T C is universal, then F detects I-projective objects, and it identifies I-derived functors with derived functors in the Abelian category C. Thus all our homological notions reduce to their counterparts in the Abelian category C. In order to apply this, we need to know when a functor F with kerf = I is the universal one. We develop a new, useful criterion for this purpose here, which uses partially defined adjoint functors. Our criterion shows that the universal I K -exact functor for the ideal I K KK in (45) is the K-theory functor K, considered as a functor from KK to the category Ab Z/2 c of countable Z/2-graded Abelian groups. Hence the derived functors for I K only involve Ext and Tor for Abelian groups. The derived functors that we have discussed above appear in a spectral sequence which in favourable cases computes morphism spaces in T (like KK G (A, B)) and other homological functors. This spectral sequence is a generalisation of the Adams spectral sequence in stable homotopy theory and is the main motivation for [8]. Much earlier, such spectral sequences were studied by Hans-Berndt Brinkmann

22 22 in [6]. In a sequel to this article, we plan to apply this spectral sequence to our bivariant K-theory examples. Here we only consider the much easier case where this spectral sequence degenerates to an exact sequence. This generalises the familiar Universal Coefficient Theorem for KK (A, B). 4. Homological ideals in triangulated categories After fixing some basic notation, we introduce several interesting ideals in bivariant Kasparov categories; we are going to discuss these ideals throughout this article. Then we briefly recall what a triangulated category is and introduce homological ideals. Before we begin, we should point out that the choice of ideal is important because all our homological notions depend on it. It seems to be a matter of experimentation and experience to find the right ideal for a given purpose Generalities about ideals in additive categories. All categories we consider will be additive, that is, they have a zero object and finite direct products and coproducts which agree, and the morphism spaces carry Abelian group structures such that the composition is additive in each variable ([21]). Notation 47. Let C be an additive category. We write C(A, B) for the group of morphisms A B in C, and A C to denote that A is an object of the category C. Definition 48. An ideal I in C is a family of subgroups I(A, B) C(A, B) for all A, B C such that C(C, D) I(B, C) C(A, B) I(A, D) for all A, B, C, D C. We write I 1 I 2 if I 1 (A, B) I 2 (A, B) for all A, B C. Clearly, the ideals in T form a complete lattice. The largest ideal C consists of all morphisms in C; the smallest ideal 0 contains only zero morphisms. Definition 49. Let C and C be additive categories and let F : C C be an additive functor. Its kernel kerf is the ideal in C defined by kerf(a, B) := {f C(A, B) F(f) = 0}. This should be distinguished from the kernel on objects, consisting of all objects with F(A) = 0, which is used much more frequently. This agrees with the class of kerf-contractible objects that we introduce below. Definition 50. Let I T be an ideal. Its quotient category C/I has the same objects as C and morphism groups C(A, B)/I(A, B). The quotient category is again additive, and the obvious functor F : C C/I is additive and satisfies kerf = I. Thus any ideal I C is of the form kerf for a canonical additive functor F.